e.g. y = x3 + 3x2 − 2x + 5
Cubic graphs can be drawn by finding the x and y intercepts.
Because cubic graphs do not have axes of symmetry the turning points have to be found using calculus.
Method 1: Factorisation.
If the equation is in the form y = (x − a)(x − b)(x − c) the following method should be used:
Step 1: Find the x-intercepts by putting y = 0. Step 2: Find the y-intercept by putting x = 0. Step 3: Plot the points above to sketch the cubic curve.
e.g. Sketch the graph of y = (x − 2)(x + 3)(x − 1)
Find the x-intercepts by putting y = 0.
0 = (x − 2)(x + 3)(x − 1)
x = 2 or -3 or 1
Find the y-intercepts by putting x = 0.
y = (0 − 2)(0 + 3)(0 − 1)
y = -2 x 3 x -1
y = 6
Plot the points and sketch the curve.
Note: Functions with a repeated factor have a graph which just touches the x-axis. e.g. y = (x − 2)2(x + 1)
Method 2: Transformation
The graph of the basic cubic y = x3 is shown in the diagram.
This basic cubic is moved or transformed as follows:
y = ax3 The a has the effect of changing the basic cubic in the y- direction.
It affects the steepness of the graph.
y = x3 + k
The k has the effect of moving the cubic up or down the y-axis by k units.
y = (x − h)3
The h has the effect of moving the basic cubic along the x-axis by h units.